Controlling single rare earth ion emission in an electro-optical nanocavity

Rare earth emitters enable critical quantum resources including spin qubits, single photon sources, and quantum memories. Yet, probing of single ions remains challenging due to low emission rate of their intra-4f optical transitions. One feasible approach is through Purcell-enhanced emission in optical cavities. The ability to modulate cavity-ion coupling in real-time will further elevate the capacity of such systems. Here, we demonstrate direct control of single ion emission by embedding erbium dopants in an electro-optically active photonic crystal cavity patterned from thin-film lithium niobate. Purcell factor over 170 enables single ion detection, which is verified by second-order autocorrelation measurement. Dynamic control of emission rate is realized by leveraging electro-optic tuning of resonance frequency. Using this feature, storage, and retrieval of single ion excitation is further demonstrated, without perturbing the emission characteristics. These results promise new opportunities for controllable single-photon sources and efficient spin-photon interfaces.


SUPPLEMENTARY NOTE 1. DEVICE FABRICATION
The fabrication process of our devices is described in Supplementary Fig. 1. We start with a 600 nm erbium (Er) doped lithium niobate on insulator (LNOI) film, with 2 µm silicon dioxide (SiO 2 ) as substrate. It is first thinned down to 300 nm by a reactive ion etching (RIE) process with argon (Ar) plasma. The waveguide is then defined by electron beam lithography (EBL), using hydrogen silsesquioxane (HSQ) as resist. Another Ar plasma RIE etches 180 nm into LN to form a ridge waveguide. The waveguide width is set to be 1.2 µm and the angle of etching process is ∼60°. After etching, the residue resist is removed by buffered oxide etch (BOE). The second EBL defines photonic crystal holes and the slab with HSQ. The hole dimensions are 600 nm×350 nm.
The EBL dose is carefully adjusted so that the actual dimensions of the holes match well with designed values. After that, a RIE process etches through LN. Removal of residue resist with BOE will cause small undercut in the SiO 2 layer, but the device structure is robust enough for further fabrication. The third EBL uses polymethyl methacrylate (PMMA) to lift off metal electrodes, which consist of 5 nm chromium (Cr) under 50 nm gold (Au) deposited by thermal evaporation.
The thin Cr layer is used to improve the adhesion. The gap between metal electrodes is designed to be 5 µm so that the optical mode will not be perturbed. We do not see noticeable difference in optical quality factor with or without the metal electrodes. Finally, the chip is dipped into BOE for a longer time to form a suspended structure.

SUPPLEMENTARY NOTE 2. MEASUREMENT SETUP
The schematic drawing of our measurement setup is shown in Supplementary Fig. 2. Light from a tunable laser (Santec TSL-710) is chopped by acousto-optic modulators (AOM) to generate excitation pulses. In the case of frequency sweeping, the internal piezoelectric tuning function of the laser is utilized. On-off extinction ratio of >100 dB is reached using 2 AOMs. The light is then sent to our device under test (DUT) mounted at 1 K plate of a dilution refrigerator. The reflection as well as fluorescence signal from DUT is collected via an optical circulator and sent to a superconducting nanowire single photon detector (SNSPD), sitting at 200 mK of the same fridge.

SUPPLEMENTARY NOTE 3. PURCELL ENHANCEMENT
Theoretically, the Purcell enhancement of an Er ion inside a cavity can be expressed as [1] where β = 0.22 [2] is the branching ratio of radiative transition of ErLN, n ≈ 2 is the cavity refractive index, and χ L = [(n 2 + 2)/3] 2 ≈ 2 is the local field correction. The mode volume is defined as V mode = ε|E(⃗ r)| 2 d⃗ r max(ε|E(⃗ r)| 2 ) , with integral over all space. The Purcell enhancement of an ion depends on the field strength |E(⃗ r)| at the ion location. The average Purcell enhancement of the cavity is then the average over all ions, weighted by their field intensity: Here, the effective mode volume V e f f can be expressed as This can be calculated from finite element simulation of the cavity mode profile, yielding V eff = 2 µm 3 . This gives P avg = 150.
The distribution of Purcell factor for ions in the cavity can also be extracted using these equations. We calculate the percentage of ions in the cavity that has Purcell factor larger than different P min . The results are shown in Supplementary Fig. 3.

SUPPLEMENTARY NOTE 4. SINGLE ION COUNT RATE AND g (2) ANALYSIS
Theoretically, the count rate we get from a single Er ion in the cavity can be calculated as Here, P e = 1/2 is the maximum probability an ion being in excited state after an incoherent pump.
P decay = 1 − e t/T 1 is the probability it decays to ground state during the collection window t. For us, t = 10 µs and T 1 = 10 µs, so P decay = 0.63. P cav−wg = κ ex κ ex +κ in ≈ 1/2 is the coupling rate between cavity and waveguide. P fiber−chip = 0.1 is the single side fiber-to-chip coupling efficiency. Loss in optical components such as the circulator and the fibers are included in P loss ≈ 0.6. Collection efficiency of our SNSPD is η SNSPD = 50 %. These account for the number of photons detected after each single excitation pulses. It is then multiplied by the repetition rate 1/T rep = 50 kHz to get the actual count rate. The resulting value is 236 Hz, in rough agreement with the ∼160 Hz we get from our measurement.
The photons we collect for second-order correlation measurement are from two parts, the single ion I ion and the background I bg . The background is attributed to the emission from ions weakly coupled to the cavity and the dark count from SNSPD. Here, we take g (2) ion (0) = ⟨I 2 ion ⟩/⟨I ion ⟩ 2 = 0 and g (2) bg (0) = ⟨I 2 bg ⟩/⟨I bg ⟩ 2 = 1. The signal-to-noise ratio is defined as SNR = ⟨I ion ⟩/⟨I bg ⟩. Then, the measured second-order autocorrelation function would be For us, we get g (2) (0) = 0.38. This gives SNR = 3.70, suggesting that ∼79 % of the collected photons are from a single ion. This is in agreement with our estimation of ∼160 Hz single ion count rate and ∼40 Hz background count rate. Along with the fact that the emission peaks do not split with spectral diffusion, it can be confirmed that majority of photons come from a single emitter.
Apart from device optimization to increase Purcell enhancement, improvement of g (2) (0) can mainly come from two aspects. The first is improving fiber-to-chip coupling efficiency. With optimized fiber glue process and better grating coupler design, a single-side transmission of 50 % is achievable. This will increase single ion count rate to ∼1000 Hz. The other factor lies in suppression of background Er emission. This can be done by using smaller doping concentration or tuning the cavity frequency further away. The ultimate background is the dark count from SNSPD, which is ∼20 Hz in our case. Implementing these improvements will result in g (2) (0) ≈ 0.04.

SUPPLEMENTARY NOTE 5. THEORETICAL ESTIMATION OF DC STARK SHIFT
The DC stark effect of 4 f -4 f transitions of Er ions in lithium niobate has been measured in previous literature to be 25 kHz/V·cm −1 [3] for electric field along crystal z-direction and vanish for electric field perpendicular to z-direction [4]. In our device, the tuning electric field is applied mostly along crystal y-direction so that the DC stark shift is minimized. However, the non-zero z-  voltage; c. z-component (E z ) field for 1 V DC tuning voltage. Gemeotry same with our devices is used in the simulation, where the film thickness is 300 nm, the waveguide width is 1.2 µm, the slab thickness is 120 nm, and the gap between metal electrodes is 5 µm. From Fig. 2c we can see that the electric field component E z for 1 V tuning voltage ranges from 0 at the center of the waveguide to ∼5×10 4 V/m at the edge. This will result in a non-zero DC stark shift of ∼10 MHz/V for ions that are on the edge, which also have smaller Purcell enhancement. The DC stark shift should decrease toward zero for ions in the center with larger Purcell enhancement. Still, the value 10 MHz/V is an order of magnitude smaller than the electro-optic tuning rate of ∼200 MHz/V.